How did Eratosthenes figure out Earth is round and calculated earth circumference 2000 years ago?
In: Mathematics
Grab a paper towel tube and hold it up to your eye like it’s a spyglass and you’re a pirate. Declare “yarrr!!”
Now use your new cardboard spyglass to look close to a light, but not quite. The line from your eye to the light is blocked by the wall of the tube, so the light doesn’t reach your eye. If you turn your head just a little bit then the tube aligns and you can see the light. Now you know that the tube is aligned between your eye and the light.
Eratosthenes did much the same, but with wells. These wells are long tubes that were bored straight down, where “straight down” is a direction that changes as you move from place to place.
Most of the time sunlight won’t directly shine on the bottom of a well, since the sun isn’t directly overhead. However, at 12:00 noon on certain days of the year the well and the sun align. Someone leaning over the edge of the well will see the shadow of their head against the water at the bottom of the well.
On that same day at the same time someone some distance away could try the same thing, but they find that the sun isn’t aligned. It’s a little bit to the north or south, depending on whether the second location is north or south of the place where the aligned well is. (Aside: if the second location is East or West then that also messes things up, but we’ll assume here that either the second location was due north or due south or that the second observer makes their observation at their own local noon).
At that location the observer can erect a tall, vertical pole of known height and look where the shadow of the top of that pole lands. Based on the length of the shadow and the length of the pole they know the angle of the sun.
The final piece of the puzzle is the distance between the two locations. This allows the observer to set up a ratio: “If I walk XXX miles (or stadia, as the case may be) then the sun changes angle by YYY degrees. Therefore, to make the sun change in angle by a full 360 degrees I’d have to walk <circumference of Earth> miles.”
It’s also worth noting that Erastosthenes didn’t discover that the Earth is round…people had recognized the curvature of the Earth for a very long time. Ships disappear bottom first when going beyond the horizon, which wouldn’t happen if the Earth were flat…they’d simply continue shrinking until they are too small to make out with the human eye.
Erastosthenes measured the shadow cast by two Obelisks on the same day, both at high noon. One was in Alexandria, and (with help from an assistant) the other was in Southern Egypt. If the Earth was flat, the shadows would be the same length, due to having the same angle relative to the sun. Instead, the lengths were different, and because the distance between the two cities were known, Erastosthenes could use geometry to estimate the difference in angles between the two obelisks, relative to each other. From there, he could use geometry to calculate the distance necessary to achieve 360 degrees, a circle.
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